WASHINGTON (CNN) - The Senate should wrap up work on a sweeping overhaul of U.S. immigration laws before the Fourth of July, but its odds of passage remain uncertain, the chamber's Republican leader said Sunday.

Despite an agreement last week that revived the White House-backed measure, "It's hard to know whether the votes will be there to pass it or not," Minority Leader Mitch McConnell told CBS' "Face the Nation."

McConnell predicted the Senate would wrap up work on the measure "one way or the other" before the holiday.

Senate Majority Leader Harry Reid, D-Nev., yanked the bill off the floor June 7 after supporters failed to muster the votes necessary to cut off debate on a series of amendments that were nibbling away at the delicate political balance the bill's authors had crafted. But Senate leaders agreed to bring the measure back after reaching a deal to consider about 20 amendments, split evenly between Democratic and Republican proposals.

The measure would create a guest-worker program to let migrant workers from other countries work temporarily in the United States, a plan that critics have said would create a permanent underclass of poor, low-skill workers.

But the bill's most controversial aspect is the creation of a pathway to legal status and eventual citizenship for the estimated 12 million illegal immigrants already in the country - a plan critics denounce as "amnesty."

Bush has made reforming U.S. immigration law a top priority for his second term, and Thursday's breakthrough came after he backed an amendment that would that would use $4.4 billion in fees raised by the legislation to boost border security and prevent illegal immigrants from being hired in workplaces.

The bill drew support from just seven Republicans on the procedural motion that led to it being shelved, along with 37 Democrats and one independent - a total of 15 votes shy of the 60 needed to move forward. Reid said Bush needed to bring more GOP allies around to supporting the bill, and Bush's proposal was an effort to woo critics who say the bill needs to place more emphasis on border security.

But the plan still has its critics in both houses of Congress, and Rep. Duncan Hunter, R-Calif., told CNN's "Late Edition with Wolf Blitzer" that the Bush-backed amendment is "a terrible trade."

"Border security is the obligation of the American government," said Hunter, a GOP presidential candidate. "That's like saying we'll send enough bullets to our troops in the field in Iraq or Afghanistan if you do something else, if you in Congress will make the right move. That should be a given."

Sen. John Cornyn, R-Texas, called Bush's plan "a good idea." But he added, "That alone is not going to satisfy the concerns about whether we are really going to build a workable system."

"Part of the problem is the American people look at this and they remember what happened in '86, when they were told, if you'll accept a one-time amnesty, then we'll get true enforcement," he told CNN. "Well, we all know what happened. We got an amnesty, but no enforcement."

But Sen. Lindsey Graham, R-S.C., said it would be "political malpractice" to simply focus on current law without addressing the status of the immigrants now in the United States illegally.

"This is no longer about immigration reform. This is about, can we govern ourselves?" Graham said. "Can Republicans and Democrats sit down at a table and do the hard things, or are those days behind us? I am confident that the Senate will deliver."

WASHINGTON (CNN) - A Capitol Hill house where Democratic presidential hopeful Sen. Barack Obama rents an apartment caught fire Sunday, but the blaze was extinguished quickly, a fire official said.

Obama, of Illinois, was with his family in Chicago on Sunday, a campaign spokesman said. No one was in the house at the time, no one was hurt and the damage was not extensive, District of Columbia fire Lt. Paul Wood said.

Wood said the fire broke out in the middle of the afternoon. It was contained between the third floor and the roof of the building, in northeast Washington, he said.

Robert Gibbs, a spokesman for the candidate, said the fire was likely started by a malfunctioning fan. He said Obama rents a second-floor apartment in the building.

WASHINGTON (CNN) - Presidential Candidate Duncan Hunter thinks the President’s new plan to provide additional border security funds to the immigration bill is “terrible”. Speaking on CNN’s Late Edition with Wolf Blitzer, Hunter said, “I think it's a terrible trade for the president of the United States or the Senate, which really has this amendment, to trade border security for amnesty.”

The president Thursday proposed adding an amendment to the immigration bill that would immediately provide $4.4 million to beef up border security. “Border security is the obligation of the American government,” Hunter said. “That's like saying we'll send enough bullets to our troops in the field in Iraq or Afghanistan if you do something else, if you in Congress will make the right move. That should be a given.”

(CNN) – Delaware Senator Joseph Biden took issue Sunday with his senate colleague, Joseph Lieberman, over how best to deal with Iran. Lieberman, of Connecticut, says there is evidence Iran is training and arming insurgents who are killing Americans in Iraq. Last week he suggested the U.S. should be prepared to take military action against Iran to stop such attacks. Biden clearly disagrees.

"Now if Joe could come up with me and tell me how he's going to, not even being able to quell things in Iraq, how he's going to go into a country of 72 million people, attack and not have our problems metastasize well beyond what they are now, I'd like to know what that is." Biden, a Democratic candidate for president made the comments Sunday on ABC's "This Week."

"We should be using a lot more imaginative diplomacy. We should be engaged in isolating Iran. We should be engaged in attracting the support of the Iranian people," Biden went on to say.

On the race for the Democratic nomination, Biden did not hesitate when asked if he, or Senator Hillary Clinton, were better qualified to be commander in chief. "Me, by a long shot. I think she's a very qualified person, but I've been doing this for 34 years of my life." Also a candidate for president in 1988, Biden said the next president will have virtually no room for error. "And I'm the only one that's even put together a plan on Iraq."

Biden has proposed partitioning Iraq into three separate regions – Kurdish, Sunni, and Shiite – with a central government in Baghdad.